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Graphing:
Absolute Value Inequalities
Notes
Absolute Value Inequalities have a lot in common with Absolute Value Equations (as well as linear equations and inequalities).
Here you can see the difference is that:
equations use an = inequalities use <, >, <, or >
k k hxayhxay
For Both:a – widens, narrows, of flips the graphh – moves graph left ( + ) or right ( - ) (*opposite of sign)k – moves graph up ( + ) or down ( - ) (*same as sign)
Absolute Value Inequalities have a lot in common with Absolute Value Equations (as well as linear equations and inequalities).
Here you can see the difference is that:
equations use an = inequalities use <, >, <, or >
k k hxayhxay
For Both:a – widens, narrows, of flips the graphh – moves graph left ( + ) or right ( - ) (*opposite of sign)k – moves graph up ( + ) or down ( - ) (*same as sign)
But we need to use the same “rules” for the Absolute Value Inequalities as we did for Linear Inequalities as far as the lines and shading are concerned.
k k hxayhxay
The graph for both of these always “start out as” the graph of:
Which looks like this:
*and is basically thetop half of the graphs of: y = x and y = -x
For y=|x|, a=1, h=0, and k=0.
and when you change the a, h, and k you are just moving the graph around the plane buy the vertex.
xy
k k hxayhxay
You need this, write this down
Variable Condition Result
a a > 1 Graph narrows, gets steeper (*opposite of what you think)
a < 1 Graph widens, gets flatter (*opposite of what you think)
- a Graph opens down
h - h Graph moves right (*opposite of what you think)
+ h Graph moves left (*opposite of what you think)
k - k Graph moves down
+ k Graph moves up
123 xyVariable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
y = |x| (Parent function)
123 xy
First, a = 3, so I narrow (steepen) the graph accordingly.
Variable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
y = |x| (Parent function)
123 xy
First, a = 3, so I narrow (steepen) the graph accordingly.
Second, h = -2, so I take the vertex and move it 2 units right.
Variable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
y = |x| (Parent function)
123 xy
First, a = 3, so I narrow (steepen) the graph accordingly.
Second, h = -2, so I take the vertex and move it 2 units right.
Third, k = 1, so I take the vertex and move it 1 units up.
Variable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
y = |x| (Parent function)
123 xy
Fourth, the sign is < so I know the lines need to be solid, and I adjust my lines.
Variable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
y = |x| (Parent function)
123 xy
Finally, the sign is < so I know I need to shade below.
Fourth, the sign is < so I know the lines need to be solid, and I adjust my lines.
Variable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
123 xyVariable Condition Result
a a > 1 Graph narrows, gets steeper
a < 1 Graph widens, gets flatter
- a Graph opens down
h - h Graph moves right
+ h Graph moves left
k - k Graph moves down
+ k Graph moves up
